## Trigonometric Ratios This topic is a review of Pythagoras' Theorem and Trigonometric Ratios covered in earlier years.

### Pythagoras Theorem

The Theorem of Pythagoras states that in a right-angled triangle the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the two other sides (the two shorter sides). h2 = a2 + b2 The hypotenuse, h, is always opposite the right angle. The converse of the theorem is also true. i.e. If h2 = a2 + b2 , then the triangle is right-angled. Examples Answers Find the length of the side marked h  Find the length of the side marked x  ### Trigonometric Ratios

There are several methods used for finding angles and sides in right-angled triangles. You may have been taught and use a different one to that shown below. They all produce the same answers if done correctly!

For a right-angled triangle: y is the side opposite angle α. x is the side adjacent to angle α. r is the hypotenuse.

 REMEMBER Sine of angle α = SOH Cosine of angle α = CAH Tangent of angle α = TOA

These can be rearranged to:
 y = r sin α x = r cos α y = x tan α

The ratios can be found using a calculator or from a book of trigonometric tables. Take care to ensure the correct mode is selected when using a calculator DEG or RAD

The trigonometric ratios are also known as circular functions and the sine, cosine and tangent of any angle positive or negative can be found.

Click here for an activity showing the sine as a circular function.
Click here for an activity showing the cosine as a circular function.
Click here for an activity showing the tangent as a circular function.

### Solution of Right-angled Triangles

The sides and angles of right-angled triangles can be found using the trigonometric ratios. For each problem the information given should be written down and substituted into one of the ratios, which can then be solved as an equation.

Remember to use Pythagoras' Theorem if only sides are involved. The final answer should be rounded off to a similar degree of accuracy to that given to measurements in the question.

 Examples Answers Find the length of x  On a calculator the step of writing out cos 50° can be missed out:      5.142300878 which rounds to 5.1 Find angle   On a calculator this last step is done as follows:       36.86989765 which rounds to 36.9°